Re: Sets vs. categories as a foundation

"Shmuel (Seymour J.) Metz" <[EMAIL PROTECTED]> writes:

 : In <[EMAIL PROTECTED]>, on 11/28/2003
 :    at 12:50 PM, [EMAIL PROTECTED] (George Greene) said:
 : 
 : >This is a very set-centric view.
 : >I thought the whole purpose of bothering with
 : >category theory in the first place was to escape this view.
 : 
 : No. Category theory is a tool for other branches of Mathematics,
 : whether you take categories as fundamental or take sets as
 : fundamental.

ANYthing that you take as fundamental, purely BY VIRTUE of
the fact that you have taken it as fundamental, becomes
MORE THAN JUST "a tool for other branches of mathematics".
If your attempt take it as fundamental has actually succeeded,
then it is, BY DEFINITION of "fundamental", a tool for ALL other
branches of mathematics.  This thread was started by the allegation
that all categories are in some deep sense imitative of the category
of sets.  That is, I REiterate, a very set-centric view.  It is a view
of sets as the fundamentally deeper concept, as somehow unavoidable,
EVEN when you are doing something explicitly designed to avoid them
(i.e. attempting to take something *else* as "fundamental", something
like categories).

 : >My point is simply that if you have ZFC, what do
 : >you need categories for? 
 : 
 : Because you can prove things once in Category Theory and then apply
 : them to various branches of Mathematics. Your question is like asking
 : what you need groups for.

No, it isn't.  There are a lot of things you CAN'T DO with groups.
Groups are unusually defined and limiting.  That's why you need
semigroups, quasigroups, loops, and algebras.  Groups don't even PRETEND,
don't even ASPIRE,  to be able to do it ALL.  But sets and categories
DO.  So my question is NOT like asking that.

 : >"The category of sets" starts to get viciously circular. 

 : Why is that an issue?

Because the vicious circle principle is a matter of basic
intellectual hygiene, that's why.  Also because circular
definitions risk sheer irrelevance and meaninglessness.

 : You can do Set Theory without the Axiom of Foundation.

Sure, but that's Work.  It TAKES Work to make sure that your
circles aren't vicious.  If you are going to talk about a
"category of all sets" as something SIMPLE (as OPPOSED to
as something as COMPLEX as a topos) then that work is going
to become overwhelming.

My point is: a) if sets are foundational then "the category of
all sets" is just irrelevantly circular.   There are a whole lot
of other NON-categorical set-structures in that world and they
are every bit as important as the categorical ones.   To choose
to focus on categories at all is to throw THOSE structures
OUT of the universe of discourse.   Saying that you have somehow
gotten them all back in because you mean the category of "all"
sets is just hubris. b) if categories are foundational and
sets are not, then the whole question of how to even define
a set is COMPLICATED -- it is at least as complicated as the topos
axioms.  It therefore becomes hard to maintain that *simpler*
categories -- ones that  that come nowhere near topoi -- are
*still* imitative of the category of all sets.  How, for example,
could any category with a finite number of objects be imitative
of one with an infinite number?  And why is it even meaningful to
speak of ONE category of all sets when THE DISCRETE category of
all sets is ALSO a category of all sets?  I mean, there is more
than 1 category whose object-domain is the class of all sets (or
the class of all identity-functions from a set to the same set).
Or are you just going to allege that any set encoding a surjective
function on one set onto another is automatically an arrow from
the one to the other, regardless of whether any of them likes it
or not?  If so, then, since that representation-of-a-function-as-a-set
is itself a set, there are going to be unavoidable arrows from it
(arrows from arrows to arrows, as OPPOSED to from&to objects).

This is all simply trying to say that a category of all sets,
whether with topoi or without, must necessarily be a complicated 
thing.  Given that there are simple categories out there, they
should not be alleged to be participating, not even by analogy,
in all that -- IRRELEVANT, FOR THEM -- complexity.
-- 
 ---
  "It's difficult ... you need to be united to have any 
   strength, but internal issues have to  be addressed."
 --- E. Ray Lewis, on  liberalism in America